I'll assume the military path is not your choice.
So...
I would finish out the MS and start working in the engineering field. You don't want to lose the academic momentum you have now and it will obviously help you out in your engineering job search (as well as help out in the salary dept).
Then, I would start my flight training at a local flight school. Pick a school with a good rep. and one which can carry you from PVT to COM/CFI. You can pay as you go while flight training and thus keep yourself out of any real debt (can't emphasize this point enough).

Arrange with the school, when you start your training, to become a part-time CFI once that rating is attained. Have at least a gentleman's agreement on it.

Once you start working as a CFI you can build your hours while using your "day job" to pay the bills and maintain a decent lifestyle.

Once you have a few hundred hours under your belt, you can then reeval. your career options.

So:
1) Finish grad school
2) Get a good paying engineering job
3) Begin your flight training locally
4) Pay-as-you-go (no major amount upfront - a lot has happened over the years)
5) Get your CFI and start instructing part-time
6) Once all that smoke clears figure out what you'd rather do once you reach ~1000 hours or so.

Hope this helps...